Algebraic generators of sequences for communication signals

ABSTRACT

A device for modulating communication signals comprises a transceiver for receiving and transmitting the signal, a storage medium storing computer implemented programme code components to generate sequences and a processor in communication with the storage medium and transceiver. The processor executes computer implemented programme code components to generate a family of shift sequences or arrays using exponential, logarithmic or index functions and a polynomial or a rational function polynomial in t∈   p−1  for a finite field    p  of prime p. Multiple columns of the arrays are substituted with pseudo-noise sequences or other suitable good correlation sequences in a cyclic shift equal to the shift sequence for the respective column to generate a substituted array. The substituted array, or a sequence unfolded using the CRT from the array when the array dimensions are relatively prime, is applied to a carrier wave of the communication signal to generate a modulated communication signal.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No.14/125,766, filed Dec. 12, 2013, which is a 371 of InternationalApplication No. PCT/AU2012/001473, filed Dec. 3, 2012, which claimspriority to Australian Application No. 2011905002, filed Dec. 1, 2011,the entire contents of each of which are incorporated herein byreference.

FIELD OF THE INVENTION

The present invention relates to generation of communication signals. Inparticular, the present invention relates to improved sequences forencoding wireless and optical communications. In particular, but notexclusively, embodiments of the invention relate to frequency hoppingand CDMA encoding methods. The improved sequences also have applicationsin radar, sonar, GPS, ultrasound and cryptography, but are not limitedto such applications.

BACKGROUND OF THE INVENTION

Typical radio communications systems, such as cellular voice and datacommunications systems, have multiple base stations in various locationsaccessed by mobile or fixed user terminals, such as cellular telephonesor wireless web devices. Each base station communicates with a userterminal using a communications channel. The communications channel canbe used for sending signals that communicate information, such as userdata and/or control data.

A communications channel may consist of transmissions in a set of timeslots in a Time Hopping Frame, or in a set of tones in a FrequencyHopping Frame, or in a bi-phase or polyphase sequence in a Code DivisionMultiple Access (CDMA) Frame. The transmissions are all on a physicalcarrier frequency. A physical carrier frequency may be a 625 kHz bandaround a central frequency, such as 800 MHz or 1.9 GHz, or an arbitrarywidth band around 1 GHz. The same principles apply to radar and sonar,except that the information is in the properties of the channel.

FIG. 1 shows a typical implementation of a modern day CDMA transceiver.The digital operations are typically implemented in a Field ProgrammableGate Array (FPGA), possibly in conjunction with a microprocessor. Userdata is modulated by a special binary or n-ary sequence, which isconverted into phases of a carrier around 1 GHz by a process calledDirect Digital Synthesis. This consists of looking up a sine/cosinetable at a fixed clocking rate, thereby generating a modulatedsubcarrier at tens of MHz. The subcarrier is generated in two versions,I (in-phase) and Q (quadrature), so that it can be up-converted by anyhigh stability oscillator using single sideband (SSB) up-conversion.This modulated carrier is amplified, sent to an antenna and thenbroadcast.

In the receiver, the received carrier, which is modulated by the userdata and a special sequence is amplified and down-converted into digitalI and Q data streams at the subcarrier frequency. These streams aredown-converted digitally and the baseband signal converted from aCartesian basis into magnitude and phase (polar basis). The magnitude isused to control the gain of the receiver front end, whilst the phase isused as an input to a correlator. The other input to the correlator is areplica of the reference version of the special sequence. The specialsequence has the property that a periodic correlation with a timeshifted version of itself (autocorrelation) is very low, except when thetime shift is a multiple of the period of the sequence, or very close tothat. Whenever that happens, the correlation is a very large complexnumber, the magnitude of which indicates how closely the timing matches,and the phase of which indicates how closely the phase of thetransmitted carrier modified by the channel matches that of its replicain the receiver. This phase fluctuates in response to the data at thetransmitter end. In this manner, the original user data can berecovered. The deviations from ideal correlation or expected phase canbe used to track any misalignment, or channel imperfections in varioustracking loops e.g. tau dither. Other special sequences may be used tosynchronize the receiver with the transmitter from a cold start.

An issue arises if there is more than one user in the above scenario.Then, the special sequences must have low cross-correlation with eachother. Otherwise, any receiver circuits such as those described in FIG.1, could get “confused” and decode the data from an unintended user orjust produce scrambled data. Therefore, sequence sets with low off-peakautocorrelation and low cross-correlation are required for a multipleuser environment. Binary (two phase) sequence sets are particularlyprized for this application. The best known of these are Gold sequencesand the large and small Kasami sequences. These are available forspecial lengths and have fixed family sizes, sequence balance, andresistance to attack. For example, Gold sequences of length 1023 areused as short, coarse acquisition codes in GPS.

The proliferation, acceptance and reliance upon personal mobilecommunications has created the need for flexible, mobile networksranging from femtocells typically used in homes and small businesses toregular cells covering larger areas. However, traditional methods ofmultiplexing multiple users by frequency or time division areinefficient and ineffective in guarding against multi-path and mutualinterference, noise and jamming or other attacks.

Code division multiple access (CDMA), frequency hopping, orthogonalfrequency-division multiplexing (OFDM), and ultra wide band (UWB)techniques have been developed in an effort to overcome theseshortcomings. These techniques have recently been enhanced with the useof Multiple Input, Multiple Output (MIMO), polarization and space-timecoding. All these techniques rely on sets of digital sequences with lowoff-peak autocorrelation and low cross-correlation. For fixedcommunication networks, it may be possible to coordinate the signals, sothat these low auto- and cross-correlation values only need to beachieved for a specified set of delays. However, due to the mobility ofthe handsets, the mobile-to-base links cannot be precisely coordinatedand require some means of separating the different users within the samechannel.

One way of achieving this is by modulating the phase of the carriersignal of each user by a unique signature sequence to achieve CDMA.Since such sequences spread the spectral occupancy of each user they arealso called spreading codes in spread spectrum terminology. It ismathematically impossible to create signature sequences that areorthogonal for arbitrary starting points and which make full use of thecode space. Therefore, unique “pseudo-random” or “pseudo-noise” (PN)sequences are used in asynchronous CDMA systems. A PN code is a binarysequence that appears random, but can be reproduced in a deterministicmanner by intended receivers. These PN codes are used to encode anddecode a user's signal in asynchronous CDMA. PN sequences arestatistically uncorrelated and the sum of a large number of PN sequencesresults in multiple access interference (MAI) that is approximated by aGaussian noise process due to the central limit theorem.

Gold codes and Kasami codes are examples of a PN code suitable for thispurpose because there is low correlation between the codes. However, ifall of the users are received with the same power level, then thevariance (e.g., the noise power) of the MAI increases in directproportion to the number of users. The signals of other users willappear as noise to the signal of interest and interfere slightly withthe desired signal in proportion to number of users.

Gold codes are a family of sequences based on binary addition ofpreferred pairs of maximal length sequences (m-sequences). Codes existfor lengths 2^(n)−1, n≠0 mod 4. Each family has 2^(n)+I codes, theoff-peak autocorrelation and cross-correlation of which is three valued.For n odd, the values are optimal.

Kasami sequences exist for lengths of the type 2^(n)−1, n even. Thereare two types: a small set and a large set. The small set is based onthe binary addition of preferred pairs of sequences, one of which is oflength 2^(n)−1, whilst the other is of length 2^(n/2)−1. The large setis formed by binary addition of triples of suitably chosen sequences oflength 2^(n)−1.

Other constructions of CDMA sequences include No Kumar, Kerdock and BentSequences. All known constructions are available for only a few selectedlengths, many are unbalanced, and all have normalized linearcomplexities which tend to 0 as sequence length increases.

For frequency hopping systems, each user is assigned a frequency hoppingcode of period T, where a single tone or a combination of tones istransmitted during each time interval, or chip period t. The lowoff-peak autocorrelation and low cross-correlation of the codes ensuresthat the data sent to or from each user suffers low and constrainedinterference, which is tolerable, or corrected for by standard errorcorrection techniques. This is also true for UWB systems, where timehopping instead of frequency hopping is involved. Similar methods arealso used in optical fiber communications, where wavelength selection isequivalent to frequency assignment and the codes are known as OpticalOrthogonal Codes (OOC).

Existing frequency hopping codes include Costas arrays which are doublyperiodic arrays with one dot per row and column, such that all thevectors separating the dots are unique. The off-peak periodic dotautocorrelation (auto hit) of such arrays is exactly 1. Generally, sucharrays are solitary, although pairs of arrays with dot cross-correlation(cross hit) constrained by 2 are possible.

Hyperbolic sequences are another example of existing frequency hoppingcodes. For Finite Field GF(p) these are defined as:

y(k)=α/k(mod p)k=1,2, . . . ,p−1 and some α≠0∈GF(p)

Hyperbolic sequences of length p−1 have at most two hits in itsauto-correlation function except for the (0,0) shift, and at most twohits in its 2D cross-correlation function formed with any other codefrom the same family.

An (n, ω, λ) Optical Orthogonal Code (OOC) C where 1≦λ≦ω≦n, is a familyof {0,1} sequences of length n and Hamming weight ω satisfying:

Σ_(k=0) ^(n-1) x(k)y(k⊕ _(n)τ)≦λ whenever either x≠y or τ≠0.

λ is the maximum correlation parameter. A review of known constructionsis presented in O. Moreno, R. Omrani, and S. V. Maric, “A NewConstruction of Multiple Target Sonar and Extended Costas Arrays withPerfect Correlation” 40th Annual Conference on Information Sciences andSystems, 22-24 Mar. 2006, pp. 512-517.

Regarding existing multi-target codes, by definition the number of hitsin the 2D auto-correlation function of Costas arrays is ideal. However,the multi-user environment and rather poor 2D cross-correlationproperties of Costas arrays necessitated the construction of othersequence families such as nth order congruence codes. GF(p) denotes theFinite Field over a prime p, and GF _((p))=GF(p)−{0}. Therefore, the nthorder congruence sequence of length p is defined by:

y(k)=ak ^(n)(mod p)0≦k≦p and some α∈

(p)

An nth order congruence sequence constructed over the Field of prime pis a full sequence when d=(φ(p),n)=1 where φ denotes the Euler φfunction and d=(x,y) is the greatest common divisor of x and y. An nthorder congruence sequence (n>1) of length p has at most n−1 hits in its2D auto-correlation function except for the (0,0) shift, and at most nhits in its 2D cross-correlation function formed with any other sequencefrom the same family.

All of the methods described above can also be used in radar, where theencoded transmitted signal is reflected or scattered by the environmentand is received at a receiver co-located with the transmitter(monostatic) or located elsewhere (bistatic). In the case of radar, theinformation is in the magnitude, phase, frequency, polarization orspatial profile of the reflected or scattered signal.

The reference to any prior art in this specification is not, and shouldnot be taken as, an acknowledgement or any form of suggestion that theprior art forms part of the common general knowledge.

OBJECT OF THE INVENTION

It is a preferred object of the present invention to provide a systemand/or a method and/or an apparatus that addresses or at leastameliorates one or more of the aforementioned problems or provides auseful alternative.

SUMMARY OF THE INVENTION

Generally, the present invention relates to methods and apparatus forencoding communication signals for applications such as, but not limitedto mobile communications, radar, sonar, GPS, ultrasound andcryptography. Embodiments of the methods include constructing familiesof shift sequences or arrays using polynomials and using the shiftsequences as frequency/wavelength or time hopping patterns for digitalcommunications. Methods include converting the shift sequences, such assubstituting columns of frequency hopping arrays with pseudo randomnoise sequences and applying the resulting arrays to a range ofcommunications depending on the characteristics of the resulting arrays.

According to one aspect, but not necessarily the broadest aspect, thepresent invention resides in a method of modulating a communicationsignal including:

generating a family of shift sequences or arrays using exponential,logarithmic or index functions and a polynomial or a rational functionpolynomial in i∈

_(p−1) for a finite field

_(p) of prime p;GF(p^(m)) or any function that produces a frequencyhopping pattern, and substituting multiple columns of the arrays withpseudo-noise sequences in a cyclic shift equal to the shift sequence forthe respective column to generate a substituted array; and

applying the substituted array, or a sequence unfolded from the arraywhen the array dimensions are relatively prime, to a carrier wave of thecommunication signal to generate a phase modulated communication signal.

Suitably, a shift sequence of the family has the form:s_(i)=Ag^(2i)+Bg^(i)+C where g is a primitive root (generator) of Z_(p)and i ranges from 1 to p−1, where A, B, C are elements of the base fieldZ_(p).

The method may include generating a (p−1)×p array having (p+1) columnseach of length p. Alternatively, the method may include generating ap×(p 1) array with p columns, each of length (p 1) where the indexfunction is used. Alternatively, the method may include generating ap×(p+1) array with p columns, each of length (p+1).

Suitably, the method includes generating the family of shift sequencesusing a quadratic exponential map of Construction A1 or A2.

Suitably, the method includes generating the family of shift sequencesusing a quadratic discrete logarithm map as an inverse of ConstructionA1 or A2.

The method may include generating the family of shift sequences usingfull cycles generated by a rational function map, such as Family B.

The method may include generating the family of shift sequences fromknown frequency hopping patterns, time hopping patterns or opticalorthogonal codes, or by transforming known CDMA families into shiftsequences.

The pseudo-noise sequence is over any alphabet and can be one of thefollowing: a binary or almost binary Legendre sequence; a ternary orpolyphase Legendre sequence; a binary or polyphase m-sequence; a GMWsequence; a twin prime sequence; a Hall sequence; a Sidelnikov sequence;another low off-peak auto-correlation sequence. Such pseudo-noisesequences can be computer-generated or generated recursively asdescribed herein.

The substituted array may be in the form of one of the following: abi-phase array for modulation of a CDMA signal; a multi-target trackingradar signal; a multi-target tracking sonar signal; an ultrasoundsignal; an optical orthogonal code array for modulation of an opticalCDMA signal.

The method may include substituting at least one column of the arrayswith a constant column such that the substituted array is balanced andhas symmetric auto-correlation values.

Suitably, the method includes substituting multiple rows of the arrayswith an array comprising a maximum of one dot per row.

The method may include constructing groups of families of sequences orarrays by applying invariance operations to parent arrays and unfoldingthe parent or the transformed arrays using the Chinese Remainder Theoremand assigning different groups of users different families of sequencesor arrays.

The method may include substituting blank columns in the shift sequenceto balance a CDMA sequence.

The shift sequences can be used in their own right as frequency hoppingpatterns, time hopping patterns, optical orthogonal codes or sonarsequences.

Increasing linear complexity may be achieved by using a shift sequenceand a column sequence in composition and unfolding the composition intoa long sequence using the Chinese Remainder Theorem.

The method may include generating a family of multidimensional arraysusing a composition of a family of shift sequences or frequency hoppatterns and a column sequence and modulating the phase of amultiplicity of orthogonal carriers and optionally dual polarization.

The method may include converting a column solitary sequence with lowoff-peak autocorrelation into a family of longer sequences or arrayswith low off-peak autocorrelation and cross-correlation.

The method may include unfolding a solitary array from Construction A1or A2 using a degree one polynomial using CRT, and substituting rows ina larger array used to produce Family B with commensurate row lengthwith the unfolded solitary array, resulting in a family of even largerarrays with good correlation and high linear complexity. Suitably, thesubstitution is performed as a cascade or is performed recursively.

Suitably, the family of sequences produced by using the shift sequenceto construct arrays which are then unfolded using CRT has a linearcomplexity greater than 45% of the sequence length, regardless of thesequence length.

Suitably, the shift sequence is obtained from an m-sequence, and thesubstitution column is a ternary or other non-binary pseudo-noisesequence, resulting in a new long pseudo-noise sequence obtained by CRTfrom the array.

Suitably, a family of shift sequences is obtained by a combination oftrace map and discrete logarithm from a singly or doubly periodic shiftsequence produced by an m-sequence. The family thus produced may be thesmall Kasami set, or the No-Kumar set.

The sequences may be used as error correcting codes or cryptographiccodes.

The resulting family of arrays, or their multi-dimensionalgeneralizations, can be used in watermarking or other applicationsrequiring families of arrays with cryptographic immunity.

According to another aspect, but not necessarily the broadest aspect,the present invention resides in a device for modulating a communicationsignal, the device comprising:

a transceiver for receiving and transmitting the signal;

a storage medium storing computer implemented programme code componentsto generate sequences; and

a processor in communication with the storage medium and the transceiverto execute at least some of the computer implemented programme codecomponents to cause:

generating a family of shift sequences or arrays using exponential,logarithmic or index functions and a polynomial or a rational functionpolynomial in i∈

_(p−1) for a finite field

_(r) of prime p;

substituting multiple columns of the arrays with pseudo-noise sequencesin a cyclic shift equal to the shift sequence for the respective columnto generate a substituted array; and

applying the substituted array, or a sequence unfolded from the arraywhen the array dimensions are relatively prime, to a carrier wave of thecommunication signal to generate a modulated communication signal.

Further aspects and/or features of the present invention will becomeapparent from the following detailed description.

BRIEF DESCRIPTION OF THE DRAWINGS

In order that the invention may be readily understood and put intopractical effect, reference will now be made to preferred embodiments ofthe present invention with reference to the accompanying drawings,wherein like reference numbers refer to identical elements. The drawingsare provided by way of example only, wherein:

FIG. 1 is a schematic diagram illustrating a device for modulating acommunication signal in accordance with embodiments of the presentinvention;

FIG. 2 shows a 6×7 bi-phase array generated in accordance withembodiments of the present invention, in which +1 are white cells, −1are black cells;

FIG. 3 shows an 8×7 bi-phase array generated from exponential rationalconstruction in accordance with embodiments of the present invention;

FIG. 4 shows a 6×7 optical orthogonal code array generated from anexponential quadratic in accordance with embodiments of the presentinvention;

FIG. 5 shows an 8×7 optical orthogonal code array generated using arational map in accordance with embodiments of the present invention;

FIG. 6 shows a 7×6 inverse array generated from an index function inaccordance with embodiments of the present invention;

FIG. 7 shows an inverse array based on the rational map of FIG. 4; and

FIG. 8 shows the array of FIG. 3 with columns substituted by a binary(0,1) Legendre sequence length 7.

FIG. 9 shows a 9×7 optical orthogonal code array generated fromexponential rational construction in accordance with embodiments of thepresent invention;

FIG. 10 shows an 9×7 ternary array generated from exponential rationalconstruction with one column substituted by −1, in accordance withembodiments of the present invention (shaded cells are 0's);

FIG. 11 shows an 9×7 ternary array generated from exponential rationalconstruction with one column substituted by −1 and another columnsubstituted by +1, in accordance with embodiments of the presentinvention;

FIG. 12 shows a set of known small Kasami sequences of length 63arranged in array format;

FIG. 13A shows the power spectrum of a 2.4425 GHz RF carrier phasemodulated by a known Kasami sequence of length 63; and

FIG. 13B shows the power spectrum of a 2.4425 GHz RF carrier phasemodulated by a sequence of length 63 obtained from a 9×7 array from FIG.11 in accordance with embodiments of the present invention.

Skilled addressees will appreciate that elements in the drawings areillustrated for simplicity and clarity and have not necessarily beendrawn to scale. For example, drawings may be schematic and the relativedimensions of some of the elements in the drawings may be distorted tohelp improve understanding of embodiments of the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The present invention includes a new construction of codes forapplications such as, but not limited to communications, mobilecommunications, radar, sonar, ultrasound and cryptography. Inparticular, the present invention includes a new construction offrequency hopping codes and methods of converting them into CDMA codes,which complement existing CDMA codes, and are superior in performance.One family of the new sequences is nearly optimal in terms of the Welchbound, whilst another is optimal, i.e. no better families can exist. Itshould be noted that these are the first such sequence families sinceKasami (1966) and Gold (1967) announced their constructions, and sinceWelch (1974) discovered the bound.

Performance criteria for communications, such as wireless CDMA, includethe following seven criteria:

1. Off-peak autocorrelation—a low bound is desirable;

2. Cross-correlation—a low bound is desirable;

3. The largest family size is desirable. For binary sequences of a givenlength, this is constrained by the Welch bound;

4. Code length is usually dictated by the data rate and by the requiredfamily size;

5. Alphabet. Binary is preferred. Zeros (absence of carrier) are bestavoided to maintain high efficiency;

6. Balance. The sequence should have equal number of +1's and −1's overa complete period. This is desirable so that the spectrum is flat andthe residual carrier power is low for efficiency and low probability ofintercept; and

7. Linear Complexity (LC). This is a measure of the minimum size of ashift register which could be used to generate the sequence, or thedegree of a recursion polynomial to achieve the same. This should be ashigh as possible. The Berlekamp-Massey algorithm can determine thepolynomial from 2LC terms of the sequence. Therefore, even anunsophisticated attacker can decode the sequence given 2LC termsreceived without error. Therefore, LC is a measure of the security ofthe sequence. A useful measure is the linear complexity expressed as afraction of the sequence length, i.e. Normalized Linear Complexity(NLC). In CDMA, radar, watermarking and cryptography it is desirable touse long sequences with high NLC. All known constructions of families ofsequences with low off-peak autocorrelation and low cross-correlationhave NLC which asymptotes to zero as the sequence length increases. Theconstructions in accordance with the present invention are the first toproduce families of sequences with NLC that does not asymptote to zero.In fact, many of the sequences have NLC between 0.5 and 0.9 and evenhigher, regardless of length. In watermarking applications,two-dimensional or multi-dimensional arrays are embedded in host datasuch as image, audio, or video. The methods described herein can be usedto construct arrays with high multi-dimensional complexity which can beused as watermarks. Such watermarks are more secure because an attackerwho manages to detect a part of the watermark cannot deduce the rest ofthe watermark from that information. For NLC=0.5 or greater, theattacker needs to detect the whole array.

The present invention makes use of algebraic techniques applied toFinite (Galois) Fields. Embodiments of the present invention are basedon families of patterns with low off-peak auto-correlation and lowcross-correlation.

With reference to FIG. 1, embodiments of the present invention include adevice 10 for modulating a communication signal, such as the transceiverbriefly described above. Device 10 can be implemented as a FPGA andcomprises a sequence generator 12 for generating the sequences inaccordance with the present invention as described herein. In someembodiments, a storage element such as a memory comprising a storagemedium, stores computer implemented programme code components forgenerating the sequences and a processor coupled to be in communicationwith the sequence generator 12 for executing at least some of thecomputer implemented programme code components for modulating thecommunication signal. Device 10 can comprise a low noise amplifier (LNA)14 coupled to a receiving antenna 16 to boost the received signal to anacceptable level. A local oscillator and mixer converts the signal intoa suitable intermediate frequency (IF) or zero frequency (DirectConversion). This signal is usually, but not necessarily, digitized,demodulated and correlated with a local version of the code used in thesignal transmission process in the device 10. Hence, device 10 cancomprise demodulator 18, analogue-to-digital converter (ADC) 20 on thereceiver side, correlator 22, and digital-to-analogue converter (DAC)24, modulator 26, amplifier 28 and transmitting antenna 30 on thetransmitter side. It will be appreciated that a single antenna can beshared for reception and transmission.

Execution of at least some of the computer implemented programme codecomponents includes generating a family of shift sequences or arraysusing exponential, logarithmic or index functions and a polynomial in i∈

_(p−1) for a finite field

_(p) of prime p.

Execution of at least some of the computer implemented programme codecomponents includes substituting multiple columns of the arrays withpseudo-noise sequences in a cyclic shift equal to the shift sequence forthe respective column to generate a substituted array.

Execution of at least some of the computer implemented programme codecomponents includes applying the substituted array, or a sequenceunfolded from the array when the array dimensions are relatively prime,to a carrier wave of the communication signal to generate a modulatedcommunication signal 18.

According to some embodiments, the sequence generator to produce thesequence sets of the present invention can be implemented in memory forrelatively short sequences. For example, the sequences can beconstructed off-line, using higher level packages, such as Mathematica,Maple, Matlab or Magma, and then stored in RAM and read out at asuitable clock rate. This method is limited by RAM capacity. For systemswhich already use this method of sequence generation, existing sequencescan be replaced by sequences of the present invention without changes infirmware or software. According to other embodiments, longer sequencescan be constructed “on the fly” using FPGA logic cells and shiftregisters. One block is used to construct a shift sequence and one blockto construct a column sequence. The two results are combined into anarray, which is read out sequentially as described herein.

Generation of the families of shift sequences or arrays will now bedescribed in detail followed by examples with reference to FIGS. 2-11.

A base field is chosen and a multiplicative group associated with it.For,

_(p) this can be the group

_(p)/{0}, which is of cardinality (p−1) and has a generator (primitiveroot) g. Alternatively, it can be the set of all rationals Q i.e.quotients of any pair of members of

_(p), including 0. The cardinality of this group is (p+1) and itsgenerator is a transformation mapping discovered and analyzed by O.Moreno in the 1970's. Henceforth, constructions based on this map arereferred to as “rational constructions”. Every member of such groups canbe expressed as a power of its generator. Therefore, the firstconstruction defines an exponential mapping between the integers

_(p−1) and

_(p)/{0} and the second construction between

_(p+1) and {

_(p)U∞}. An inverse (logarithmic) mapping is also possible. It is alsoenvisaged that there are other mappings which possess the requiredproperties.

In accordance with embodiments of the present invention, a family ofshift sequences or arrays is constructed using a polynomial or arational function polynomial in i∈

_(p−1), e.g. the quadratic Ag^(2i)+g^(i), where A is an arbitrarynon-zero constant. In such a family of p−1 shift sequences, the auto andcross hit values are bounded by 2. For higher degree polynomials, thebound is the degree of the polynomial. In accordance with embodiments ofthe present invention, these shift sequences can be used to encodecommunication signals. For example, the shift sequences can be used asnew frequency hopping patterns with bounded correlation.

In accordance with embodiments of the method of the present invention,the columns of the arrays can be substituted by different types ofpseudo-noise sequences in a cyclic shift equal to the value of the shiftsequence for the respective column of the array. Generally, if thesubstitution sequence is over +1 and −1, or higher roots of unity, andpossibly includes a limited number of zeros, the resulting array can beapplied to, for example, wireless CDMA. If the substitution sequence isover 1, 0 the resulting array can be used in, for example, optical CDMAor multi-tone frequency hopping.

The arrays constructed in the example described above have sides p andp−1 or p and p+1. These sizes are relatively prime and therefore thesearrays can be unfolded using the Chinese Remainder Theorem into longsequences of length p(p−1) or p(p+1).

Patent application WO 2011/050390 by the present inventors, which isincorporated herein by reference, discloses the following constructionD2 (Quadratic Generalization-Family of Arrays):

f _(A,D,C)=(i)=A(α^(i))² +Bα ^(i) +C,f:Z _(p) m ₁ →Z _(p) ^(m){0}

where A, B, C are elements of the finite field GF(p^(m)). Any two arraysin this family of arrays which are multidimensional cyclic shifts of oneanother are called equivalent. The autocorrelation of such arrays andthe cross-correlation between any non-equivalent arrays is bounded bytwo. This construction is listed in the section covering constructionsin three dimensions and higher. However, for m=1 it can be used togenerate two dimensional arrays, which can also be unfolded into onedimensional sequences.

Consider a shift sequence:

s _(i) =Ag ^(2i) +Bg ^(i) +C

where g is a primitive root of Z_(p) and i ranges from 1 to p−1. This isa shift sequence which generates a (p−1) array having (p−1) columns eachof length p.

Vertical shifts of s_(i) yield equivalent arrays (s_(i)+v≡s_(i)). Arepresentative of the vertical shift equivalence class is selected byputting C=0:

s _(i) =Ag ^(2i) +Bg ^(i)

Horizontal shifts of s_(i) yield equivalent arrays (s_(i+h)≡s_(i)):

s _(i+h) =Ag ^(2(i+h)) +Bg ^(i+h) =Ag ^(2(i+h))+(Bg ^(h))g ^(i)

As h runs through all values from 0 to p−2, g^(h) takes on all non-zerovalues in Z_(p) including the multiplicative inverse of B. Therefore, arepresentative of the equivalence class of horizontal shifts can betaken as:

s _(i) =Ag ^(2i) +g ^(i)

There are p choices of A (including A=0, the exponential Welchconstruction), so there are p arrays in the family.

Consider two such arrays s_(i)=Ag^(2i)+g^(i) and s′_(i)=A′g^(2i)+g′ andexamine the cross-correlation between them. An array produced by s_(i)is shifted horizontally by h and vertically by v:

Δ=s′ _(i) −s _(i+h) −v=A ^(rg) ^(zi) +g ^(i) −Ag ^(2(t+h)) +g ^(t+h) −v

Δ=(Ag ^(2h) −A′)g ^(2i)+(g ^(h)−1)g ^(i) −v

This is a quadratic in g^(i). Hence there are at most 2 columns whichcan match in cyclic shift between the two arrays. This is also true forauto-correlation, where A=A′. The array columns of the arrays are oflength p, so they can be substituted by binary or almost binary Legendresequences. For p=4k+1, the leading term of the Legendre sequence must be0, whilst for p−4k−1, the leading term can be set to +1. The peakautocorrelation for a column is p−1 for p=4k+1 and p for p=4k−1. Hence,the array auto-correlation and cross-correlation values are as shown inTABLE 1 below:

TABLE 1 Prime Autocorrelation Peak 0 Match 1 Match 2 Match 4k + 1 (p −1)² −p + 1 1 p + 1 4k − 1 p × (p − 1) −p + 1 2 p + 3

The arrays described above can be unfolded to yield sequences of length(p−1)×p. This is because gcd[(p−1),p]=1. Therefore, the ChineseRemainder Theorem (CRT) can be employed. The process is equivalent todiagonal unfolding of the array. Clearly, the exponential constructionis superior.

With further reference to patent application WO 2011/050390, twomatrices s_(i) produced by mapping polynomials over finite fields usinga logarithmic function are as follows:

s _(i)=log_(α)(Aα ^(2i) +Bα ^(i) +C)  Construction A1

s _(i)=log_(α)(A _(n)α^(ni) +A _(n-1)α^((n-1)i) + . . . +A _(k)α^(ki) .. . A _(o))  Construction A2

α is a primitive element of Z_(p)(p) and n is any positive integer,including 1. i is an index taking on the values 0, 1, 2, . . . , p−2.s_(i) takes on the values 0, 1, 2, . . . , p−2, ∞, where ∞ results fromthe argument of the log function being equal to 0. A, B, C and any ofthe A_(k) are suitably chosen entries from Z_(p)

1. Z_(p)={0, α¹, α², . . . , α^(p−1)}. In this context log refers tolog_(α)x=1 implies that x=—α^(j).

2. Construction A2 is a generalization of A1, where the family size islarger, but the auto and cross-correlations are also larger.

3. A matrix S is produced by placing an entry of 1 in the horizontalposition i and vertical position s_(i) and 0 elsewhere.

4. Note that the log mapping is 1:1, i.e. there is a single value of s,for each i.

The matrices from A1 have the following property: for any non-zerodoubly periodic shift of such a matrix, its auto correlation is equal toor less than 2. Some of the matrices generated are shifts of each other,and hence have bad correlation. There is an equivalence relation whichmakes (p−1)² choices of A, B or C redundant, and hence there areapproximately p inequivalent matrices in the family. It can be shownthat all inequivalent quadratics can be represented by p choices of Cin: s_(i)=log_(α) (x²+x+C).

Each of these matrices from A1 can be assigned to a different user. Adoubly periodic cross-correlation between any pair of such matrices isalso equal to or less than 2.

Columns with ∞ in them can be replaced by a string of 0's. This reducesthe peak autocorrelation by p−1, but has almost no effect on theoff-peak autocorrelation, or the cross-correlation. Where there is onlyone column with a ∞, the column can be replaced by a string of constantvalues, including +1 or −1. The autocorrelation is even better than whenthe constant is 0 whilst the cross-correlation can increase by p−1. Whenthere are two or more entries with ∞, the best option is to replace themby a string of 0's. This reduces the peak autocorrelation even further,and makes such arrays less desirable.

For Construction A1, ∞ may occur 0,1, or 2 times, depending on thechoice of A, B and C. 0 occurs if the polynomial is irreducible, 2 if itis reducible with two factors and 1 if it is a square.

Therefore it is desirable for the quadratic in A1 to be irreducible.Quadratics of the form x²+x+C yield arrays, which are not related by twodimensional cyclic shifts, and are hence inequivalent. It can be shownthat for odd p,

$C = \frac{1 - \alpha^{{2k} + 1}}{4}$

results in an irreducible quadratic, whilst for even p, a choice of Csuch that Tr_(p) ^(p) ² (C)=1 results in an irreducible quadratic.Examples of arrays generated in accordance with embodiments of thepresent invention will now be discussed with reference to FIGS. 2-8.

With reference to FIG. 2, consider a (p−1)×p array where p=7, g=3,S_(i)=g^(2i)+g^(i). This generates the 6×7 bi-phase array obtained fromthe exponential quadratic g^(2i)+g^(i). The unfolded sequence using CRTis as follows:

1,1,1,1,1, −1,1,1,1, −1,1,1,1,1,1,1,1,1, −1, −1, −1,1, −1,1, −1, −1,−1,1, −1, −1,1,1,1, −1,1, −1, −1, −1, −1, −1, −1,1.

In binary format, this translates to:

0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,1,1,1,1,1,0

The auto-correlation values are as shown in TABLE 2.

TABLE 2 Autocorrelation Value −6 +2 +10 +42 Frequency of Occurrence 1914 8 1

With reference to FIG. 2, consider a (p+1)×p array where p=7, whichgenerates the 8×7 bi-phase array obtained from the exponential rationalconstruction, called Family B: Rational Function Characteristic p. LetF_(p) _(m) be a finite field with p^(m) elements, where p is a prime.Let P be the projective line over F_(p) _(m) , in other words P is F_(p)_(m) plus cc. Consider

${f(x)} = \frac{{Ax} + B}{{Cx} + D}$

where AD≠BC and A, B, C, D∈F_(p) _(m) . The substitution of elements ofP in f(x) produces a permutation of the elements of F. If g(x) isanother fractional linear transformation, similar to f(x), then thereare exactly two values of x in P for which f(x)=g(x). The followingresult is due to Berlekamp and Moreno: Whenever x²+x+α is irreducible,and a is primitive in F, then the permutation given by

${C(x)} = \frac{- \underset{\_}{\alpha}}{x + 1}$

gives a cycle or length p^(m)+1. It is assumed now that the cycle givenby

${C(x)} = {- \frac{\alpha}{x + 1}}$

begins with 0 and ends with ∞, and since 0→−α, it goes 0→−α, . . . , →∞.

The eighth column would normally be left blank, but in this embodiment,the method includes substituting the eighth column with a constantcolumn, so that the resultant array is balanced and has symmetricautocorrelation values. The auto-correlation values are as shown inTABLE 3:

TABLE 3 Autocorrelation Value +56 +8 0 −B Frequency of Occurrence 1 1616 23

The array can be unfolded using CRT into a sequence, binaryrepresentation of which is:

1,0,1,1,1,1,0,1,1,1,0,0,1,1,0,1,1,1,0,1,0,0,0,0,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,1,1,1,0.

A double periodic shift sequence of length (q+1)(q−1) can also beconstructed from a cycle generated by a rational function modulo q−1.q+1 and q−1 are relatively prime when q is a power of 2. Suchconstructions have two blank entries, compared with those of the type(q+1)q. FIGS. 9, 10 and 11 show examples of this construction. The twoblanks offer several options of substitution for the purposes of goodbalance. By contrast, known constructions such as Gold and Kasami havepoor balance. For primes of the type 2^(n)−1, binary m-sequences can besubstituted. For a few suitable primes of the form 4k²+27, binary Hallsequences can be substituted. For polynomials of degree d and primes ofthe type 4k−1, where a binary Legendre column substitution is used, theoff-peak autocorrelation and cross-correlation are bounded by[(d−1)p+d−1], whilst the peak is p(p+1). For primes of the type 4k+1 theonly substitution possible is a ternary Legendre column, so the off-peakautocorrelation and cross-correlation are bounded by [(d−1)(p−1)+d] andthe peak is (p−1)².

For comparison of known codes with the codes generated in embodiments ofthe present invention, for example, the family of Gold codes of length63 has off-peak autocorrelation and cross-correlation values of −17, −1and +15. In general, the performance of sequence families depends ontheir length. Therefore, it makes sense to normalize the parameters andto compare sequences of similar length. The Linear Complexity (LC) ofthe sequences involved in the present invention up to length 25,000 wastested by computer using the Berlekamp Massey algorithm.

The inventors have developed a theory which yields an estimate of thelinear complexity for the new sequences. TABLE 4 below shows theperformance of various constructions produced by embodiments of thepresent invention compared to the existing constructions. It is clearthat many new lengths are added and, in general, the new sequences aresuperior. R refers to rational map, B refers to balanced Legendrecolumn. The sequences according to embodiments of the present inventionare available in many more lengths than traditional sequences, as seenin Table 4.

TABLE 4 Normalized Balanced Total Efficiency Correlation Family FamilyNormalized Type Length % Bound Size Size Complexity Gold 31 100 0.29 1733 0.32 Moreno Tirkel 2 (p = 7) 42 100 0.238 0 48 0.453 Moreno Tirkel 2B(p = 7) 42 85.7 0.222 48 48 >0.5* Moreno Tirkel 2R (p = 7) 56 100 0.14348 48 0.768 Gold 63 100 0.27 33 65 0.19 Moreno Tirkel 2 (p = 1l) 110 1000.127 0 120 ≈0.455 Moreno Tirkel 2B (p = 11) 110 81.8 0.156 120120 >0.5* Gold 127 100 0.13 65 129 0.11 Moreno Tirkel 2R (p = 11) 132100 0.091 120 120 >0.5* Moreno Tirkel 2 (p = 13) 156 92.3 0.097 168 168≈1 Moreno Tirkel 2R (p = 13) 182 85.7 0.095 168 168 ≈1 Small Kasami 255100 0.067 0 16 0.047 Large Kasami 255 100 0.129 0 4111 0.062 MorenoTirkel 2 (p = 17) 272 94.1 0.07 288 288 ≈1 Moreno Tirkel 2R (p = 17) 30688.9 0.063 288 288 ≈1 Moreno Tirkel 2 (p = 19) 342 100 0.064 0 360 >0.5*Moreno Tirkel 2B (p = 19) 342 89.5 0.072 360 360 >0.5* Moreno Tirkel 2R(p = 19) 380 100 0.053 360 360 >0.5* Moreno Tirkel 3R (p = 19) 380 1000.105 6858 6858 >0.5*

A comparison of the asymptotic behavior of various known sequencefamilies with two of the sequence families of the present invention as Ltends to infinity is shown in TABLE 5. The general form of the sequencesgenerated by the present invention, which are derived from higher degreepolynomials, is expected to be very useful to a CDMA designer, since itallows for a flexible number of users. The table shows sequences derivedfrom polynomials of degree 2 and 3 only.

TABLE 5 Normalized Balanced Total Correlation Family Family NormalizedType Bound Size Size Complexity Gold L = 2^(n) − 1 n odd L^(−0.5) L/2 L0 Gold L = 2^(n) − 1 n 2L^(−0.5) L/2 L 0 even Small Kasami L^(−0.5) 0L^(0.5) 0 Large Kasami 2L^(−0.5) 0 L^(1.5) 0 Moreno Tirkel L^(−0.5) LL >0.5 degree 2 Moreno Tirkel 2L^(−0.5) L^(1.5) L^(1.5) >0.5 degree 3

Sequences can be generated mathematically by recursion polynomials, orpractically by linear shift registers, which are a physical embodimentof these recursion polynomials. The inventors have established that therecursion polynomials for sequences in the exponential quadratic familyhave the same structure as the recursion polynomials of the columnsequence used in the construction. The long sequence polynomial isobtained by raising the terms in the column sequence polynomial to thepower (p−1). This theory has been verified by computer for sequencelengths up to 25,000. The sequences in the families constructed usingthe rational function map are more complicated, because the columns inthe parent array are of two types, so that other terms appear in theirrecursion polynomials.

The theory has also been verified for ternary column sequences and fordifferent substitutions for blanks in the shift sequence. By using theLegendre sequence as a column sequence, normalized complexities of atleast 0.5 can be obtained for all sequence families of the presentinvention. This is regardless of sequence length. This is the first timesuch asymptotic performance has been achieved. This can be deduced bytheoretical means and has been verified experimentally by the inventors.In fact, the inventors have developed an algebraic method of computingthe linear complexity of the long sequence unfolded using CRT and eventhe recursion polynomial to generate the long sequence.

The Berlekamp-Massey algorithm is a universally accepted means ofanalysing linear complexity, and hence the most likely tool of anattacker. The output of the Berlekamp-Massey algorithm is the recursionpolynomial of the sequence under test. In order to compute the recursionpolynomial, the Berlekamp-Massey algorithm requires 2LC terms of thesequence to be known without error. LC is the linear complexity of thesequence. Since our sequences have normalized complexities exceeding0.5, this implies that an attacker must have access to at least a singlerepeat of our sequence in order to decode it. In most spread spectrumapplications, the user data modulates the spreading sequence, at arepetition rate of the sequence period. Therefore, an attacker has noguarantee of analysing more than a sequence period, without sequencecorruption by data modulation. In the case of watermarking, the attackerhas access to one period only, and has to guess how to synthesizerepeats, without knowing the start and end of a sequence. In summary,the new sequences are practically immune to linear attack, even when thesequence is not corrupted by noise, interference or distortion.

The sequences/arrays described above with reference to FIGS. 2 and 3 canbe applied as a bi-phase modulation to a sine wave carrier of acommunication signal. The sequences can be used in classical wirelessCDMA signals, multi-target tracking radar and sonar signals, medicalultrasound signals, and neonatal audio testing, and the arrays can beused in digital watermarking.

In other embodiments, an alternative array construction is based on theuse of the index function, which is an inverse of the exponentialfunction. This is different from the log function, which applies to allfinite fields. The index function is restricted to base fields only. Theshift sequence is as follows:

s _(x) =ind _(g)(Ax ² +Bx+C)

Here x∈Z_(p) (including 0) and g is the primitive root (generator). Thisgenerates a p×(p−1) array with p columns, each of length (p−1).

A vertical shift by v generates an equivalent array:

s_(x) ≡ s_(x) + v = ind_(g)(Ax² + Bx + C) + v = ind_(g)(Ax² + Bx + C) + ind_(g)(g^(v)) = ind_(g)(Ag^(v)x² + Bg^(v)x + Cg^(v))

Choose Ag^(v)=1 from the equivalence class. Therefore, a representativeis:

s _(x) =ind _(g)(x ² Bx+C)

This array is equivalent to any shift by h places horizontally.

s _(x) ≡s _(x+h) =ind _(g)((x+h)² +B(x+h)+C)=ind _(g)(x ²+(2h+B)x+(h ²+Bh+C)

As h ranges over all values, choose B=0.

s _(x) =ind _(g)(x ² +C)

Re-express this as:

s _(x) =ind _(g)(x ² −C′)

If C′∉

, the set of quadratic residues, then (x²−C′) is irreducible, and s_(x)assumes legitimate values modulo (p−1) for all p values of x. Otherwise,blank columns result, which is undesirable. Since there are

$\frac{p - 1}{2}$

quadratic non-residues in Z_(p), there are

$\frac{p - 1}{2}$

members in the family of arrays.

The cross-correlation between two arrays is obtained from:

Δ=s′ _(i) −s _(i+h) −v=ind _(g)(x ² −C′)−ind _(g)((x+h)² −C′)−v

Let +v=λ. Then ind _(g)(x ² C′)ind _(g)((x|h)² C′)=λ.

Therefore:

${{ind}_{g}\left\lbrack \frac{\left( {x^{2} - C^{\prime}} \right)}{\left( {\left( {x + h} \right)^{2} - C^{\prime}} \right)} \right\rbrack} = \lambda$${i.e.\mspace{14mu} \frac{\left( {x^{2} - C^{\prime}} \right)}{\left( {\left( {x + h} \right)^{2} - C^{\prime}} \right)}} = g^{\lambda}$

and finally:

(x ² −C′)=g ^(λ)((x+h)² −C′)

This is a quadratic, so there may be 0, 1 or 2 columns matching, asbefore. In this embodiment, the columns of length (p−1) can besubstituted by Sidelnikov sequences of that length. Such columns haveauto-correlation values of (p−2) for a full match, and 0 or −2otherwise. It is not possible to calculate the array correlation inclosed form. However, the bounds on the correlations are:

Autocorrelation peak=p×(p−2)

Largest positive off-peak correlation=2×(p−2)

Largest negative off-peak correlation=−2×(p−2).

Therefore, this construction is inferior to the exponential constructiondiscussed before. This is because the family size is halved, thecorrelation bounds are doubled and the array efficiency is lower. Here,efficiency is the ratio of the number of non-zero entries in the arrayto the total number of entries.

Other embodiments of the present invention include the generation ofsequence families that are particularly suited to optical CDMA andfrequency hopping. FIG. 4 illustrates an optical orthogonal code (OOC)array having 6 columns and 7 rows generated from the exponentialquadratic g^(2i)+g^(i). Auto and cross-correlation are bounded by 2.Such an array can also be used in frequency hopping. The array canactually be transmitted as 6 columns and 4 rows, since the two bottomrows and the top row are empty. The empty rows can be reinserted at thereceiver before performing the correlation. In this manner, thetransmission bandwidth is reduced from 7 to 4, thus almost doubling thechannel efficiency.

FIG. 5 illustrates an optical orthogonal code (00C) array having 8columns and 7 rows generated from a rational map. Auto andcross-correlation are bounded by 2. Note, that in this case, the arrayhas one empty column out of 8. Therefore, only 7 time slots need to beassigned to the transmitter, thus increasing transmission efficiency by14%. The empty time slot can be re-inserted in the receiver, prior tothe correlation.

The arrays described above with reference to FIGS. 4 and 5 can beapplied as a frequency-time or wavelength-time pattern. The dotsrepresent the locations of transmission in the two dimensional pattern.The exponential function has inverses in the form of the index function,which applies only to the base field, and in the form of the discretelogarithm function, which applies to all finite fields. Both inversefunctions are defined for all field elements except 0.

As an example of the index function construction, an irreduciblequadratic is chosen, for example: i²+3i+1. In this example p=7:

s _(i) =ind _(g)(i ²+3i+1)=0,5,4,5,0,3,3

This defines a 7×6 inverse array having 7 columns each of length 6, asshown in FIG. 6. This array has two empty rows in the middle. However,because all of the arrays of the present invention use doubly periodiccorrelation, this array is equivalent to any doubly periodic shift ofitself. If all the rows were shifted upwards by one, the two empty rowswould be on top. Therefore, the transmitter need only send 4 out of 6tones, thus increasing channel efficiency by 50%.

A similar procedure can be applied to a rational map and an inversearray based on the rational map is shown in FIG. 7. This array containsan empty row on the bottom, so once again, the channel efficiency can beincreased by 14% by not sending the empty row.

The arrays described above in relation to FIGS. 6 and 7 and other familymembers could be used for OCDMA or for frequency hopping. Alternatively,they could be adapted for watermarking, by substituting the columns oflength 6 or 8. Suitable column substitution sequences are Sidelnikovsequences of length 6 or 8. These are preferred because, apart from aleading 0 entry, all the other entries are +1 or −1, and the sequencesare balanced and have a constant off-peak autocorrelation of −1.Alternatively, suitable sequences over a higher alphabet could be chosenas columns.

The arrays described above can be considered as sparse, in that eachcolumn contains at most one dot (transmission in the two dimensionalpattern). The arrays whose column lengths are prime can be made “denser”by substituting each column by a binary (0,1) Legendre sequence ofcommensurate length. An example of such a substitution for the arraydescribed above in relation to FIG. 4 is shown in FIG. 8 wherein thebinary (0,1) Legendre sequence is of length 7. The peak correlation forthis array is 10. This is because at most, 2 columns can match,contributing a correlation of 6, whilst the remaining 4 columns cannotmatch, contributing a correlation of 4. The array fill fraction is18/42. There are 6 such arrays, whose cross-correlation is also boundedby 10. In general, the construction in this embodiment of the presentinvention yields arrays of the type

$\left\lbrack {{\left( {{4k} - 1} \right)\left( {{4k} - 2} \right)},\frac{\left( {{4k} - 2} \right)^{2}}{2},\left( {{4k^{2}} - {4k} + 2} \right)} \right\rbrack$

for a prime of the type (4k−1). Larger arrays follow a similar pattern.Such sequence families are particularly suitable for multi-wavelengthoptical communications and multi-tone frequency hopping.

FIG. 9 shows an optical orthogonal code or frequency hopping patternwhich is obtained by using a single cycle map over {GF(2^(g))}∪{∞}followed by a permutation of the type 1/x The resulting shift sequenceis −,1,3,6,4,2,5,0,−. This array has an array off-peak autocorrelationbounded by 2. When used as an optical orthogonal code or frequencyhopping pattern, two time slots could be eliminated, as outlined above.

The columns with integer entries can be substituted by equivalent cyclicshifts of an m-sequence of length 7, producing the array in FIG. 10.Note, that in this case one of the blank columns was substituted by acolumn of −1's. This balances the array. The array can be unfolded intoa sequence of length 63, with 7 entries of 0, 28 entries of −1 and 28entries of +1. The peak autocorrelation is 56, whilst the highestoff-peak value is 9. In comparison, Gold codes exist for the samelength, but their highest correlation is only 17.

The array of FIG. 10 could have the remaining column substituted by+1's, as shown in FIG. 11. Such an array or unfolded sequence has a peakcorrelation of 63 and a worst case off-peak correlation of 11. Thissequence/array has an imbalance of 7. Both variations above in FIGS. 10and 11 outperform known Gold codes of the same length in terms ofcorrelation and linear complexity.

A larger example of this method, using the finite field GF(2⁴) resultsis a shift sequence of length 17:−,1,4,13,9,14,5,6,8,10,11,2,7,3,12,0,−. This produces an array of size17×15. It can be used as an optical orthogonal code or as a frequencyhopping sequence. Note that once again, the end columns are blank, sothey could be omitted during transmission.

The columns with integer shifts can be substituted by equivalent shiftsof an m-sequence of length 15. This array can be unfolded into a binaryor ternary sequence of length 255. Therefore, it is commensurate withthe length of the small and large Kasami sets.

It can be seen that the above shift sequence produces an array which hasat most one dot per row. Therefore, it is also possible to substitutethe rows by a pseudo-noise sequence of commensurate length. In the aboveexample, this would be a ternary Legendre sequence of length 17. Thereare other cases, where substitution of rows or columns is possible, thusleading to the construction of new and different arrays for CDMA andother applications.

In yet further embodiments, it is also possible to use the constructionsdescribed herein recursively. For example, take Construction A1 or A2where the polynomial is of degree one, i.e. the well-known exponentialWelch construction. The parent array has p−1 columns each of length p.Substitute the columns with commensurate shifts of a pseudo-noisesequence, e.g. a Legendre sequence. This array can be unfolded into asequence of length (p−1)p using CRT. Such a sequence has three valuedautocorrelation: (p−1)p, +1, −p. Quite often (p−1)p=p′+1 where p′ isanother prime. Therefore, such a sequence can be used to substitute thenon-blank rows of Family B, which are of commensurate length. Thisproduces a family of sequences of length p⁴−2p³−2p²+3p+2 with lowoff-peak autocorrelation, low cross-correlation and high linearcomplexity. Hence, the methods of the present invention can also be usedin cascade.

According to some embodiments, the method of modulating a communicationsignal includes converting singly or doubly periodic shift sequencesfrom m-sequences using a trace map and discrete logarithm to obtainfamilies of shift sequences with correlation 2. These families alsoinclude a “parent” sequence with correlation 1. Most of these shiftsequences are new, and can also be used as new frequency hopping codes,or in our constructions, where columns or rows are substituted bysuitable binary sequences to produce CDMA codes. Special cases of thismethod include the small Kasami set and the No-Kumar set. Because thesesets originate from an m-sequence, they have a normalized complexitywhich asymptotes to zero, even if their m-sequence columns aresubstituted by Legendre sequences. This is in contrast to the othersequences constructed by our method, which do not originate fromm-sequences.

According to some embodiments, where the shift sequence is obtained froman m-sequence, and the substitution column is a ternary or othernon-binary pseudo-noise sequence, a new long pseudo-noise sequenceobtained by CRT from the array results.

In some embodiments, the family of sequences produced by using the shiftsequence to construct arrays which are then unfolded using CRT has alinear complexity greater than 45% of the sequence length, regardless ofthe sequence length.

Also, it is possible to construct new shift sequences and frequencyhopping patterns from known CDMA constructions. In their paper “Familiesof sequences and arrays with good periodic correlation properties, IEEPROCEEDINGS-E, Vol. 138, No. 4, JULY 1991” D. H. Green and S. K.Amarasinghe present known Kasami sequences and their derivatives, theNo-Kumar sequences written in array format. FIG. 12 reproduces such aset of arrays from their paper. It is apparent that all these arrayscontain cyclic shifts of an m-sequence and constant columns. Thesequence of shifts, treating constant columns as blanks has auto andcross-correlation bounded by 2 for the small Kasami and No-Kumar arrays.Therefore, the sequence of shifts obtained from such arrays can be usedas a shift sequence for some of the constructions of the presentinvention, and the columns can be substituted by suitable pseudo-noisesequences, such as Legendre or Hall, yielding new arrays with identicalcorrelation properties, and sometimes with much higher linear complexityand therefore immunity to attack. Additionally, sometimes, the arrayscan be unfolded into sequences and refolded in a different format,leading to still more such arrays.

Alternatively, the shift sequences can be used in their own right as newfrequency hopping patterns, time hopping patterns or Optical OrthogonalCodes.

One of the objectives of spread spectrum sequences is to provide goodspectral occupancy of the allowed frequency band, for example theunlicensed Industrial, Scientific, and Medical (ISM) band from 2.4000 to2.4835 GHz. An ideal spread spectrum sequence is pseudo-noise, i.e. thesequence possesses a two-valued autocorrelation. Because of the digitalmodulation, this produces a sinc function power spectrum. Families ofsequences used in CDMA applications have good cross-correlation, buttheir autocorrelation takes on more values than 2. As a result, theirpower spectral density is more jagged. The spectrum of a 2.44175 GHz RFcarried modulated by a typical Kasami sequence of length 63 is shown inFIG. 13A. By comparison, the spectrum of a typical Moreno-Tirkelsequence of the same length, obtained by unfolding the 9×7 array of FIG.11 is shown in FIG. 13B. Both spectra track a sinc function envelope,but the central part of the spectrum of the I sequence according to thepresent invention in FIG. 13A is slightly better. This may be due to itssuperior balance. In any case, if a system utilizing Kasami sequencesfor CDMA had its spreading sequences substituted by sequences, a slightimprovement in spectral occupancy is likely to result.

The +/−1 and (0,1) sequences described above can be appliedsimultaneously to acquire and/or track targets in a multi-targetscenario for short range multi-target radar and sonar. Hence,embodiments of the present invention include applying the shiftsequences described herein as frequency hopping sequences for radar anddetecting the presence of the shift sequences by setting a correlationthreshold of 3 for quadratic constructions and 4 for cubicconstructions.

Embodiments of the present invention also have application incryptography. The constructions generate long sequences with high linearcomplexity from shorter sequences, which can be used, for example, instream or block ciphers.

The constructions described in this patent application can be used intheir two-dimensional form in wireless communications. For example, thearrays described with reference to the drawings can be used as m×narrays, where m is the number of time slots required to carry the codedinformation and n is the number of orthogonal tones in an OrthogonalFrequency-Division Multiple Access (OFDMA) signal set. m and n need notbe relatively prime. In cases where two orthogonal polarizations aredeployed (e.g. vertical and horizontal, or left and right handcircular), the column length becomes 2n. Also, the orthogonal tones andthe polarizations need not be arranged in a one dimensional form, butcan be folded into an abstract space of arbitrary dimension. Therefore,the multi-dimensional array constructions described in the Applicants'patent application WO 2011/050390 can be used for the purposes ofmultiple access communication or radar. This applies regardless ofwhether the columns or folded arrays are substituted by suitablesequences or arrays to produce a multiplexed data stream, or whether theshift sequence is used as a selector of one out of a set of orthogonaltones and polarizations.

Hence, embodiments of the present invention include methods andapparatus for modulating and encoding communication signals that addressor at least ameliorate one or more of the aforementioned limitations ofthe prior art or at least provides a useful and effective alternative.

The new sequence families for wireless CDMA described herein havecorrelation performance which at least equals or surpasses the knownGold and Kasami sequence families. The exponential quadratic familiesare nearly optimal with respect to the Welch bound, whilst the rationalfunction families are optimal, i.e. for a given length no families withbetter correlation can exist.

The sequences come in many variations, all except one of which arebalanced, and all of which have much higher linear complexity than theknown constructions, thus rendering them immune to linear attacks andrendering them particularly suitable for cryptography applications. Themethods of the present invention include converting frequency hoppingpatterns into CDMA sequences and the converse, and deriving manypatterns from m-sequences.

The new sequences are available in more lengths, which fill in the gapsin lengths of the Gold and Kasami sequences. The new sequences do notcompete with the known sequences, but complement them, thus affordingthe user more flexibility in designing communication networks. The newconstructions also deliver frequency hopping sequences and sequences foroptical CDMA and multi-target radar and sonar, GPS and ultrasound. Inthis specification, the terms “comprise”, “comprises”, “comprising” orsimilar terms are intended to mean a non-exclusive inclusion, such thata system, method or apparatus that comprises a list of elements does notinclude those elements solely, but may well include other elements notlisted.

Throughout the specification the aim has been to describe the preferredembodiments of the invention without limiting the invention to any oneembodiment or specific collection of features. It is to be appreciatedby those of skill in the art that various modifications and changes canbe made in particular embodiments exemplified without departing from thescope of the present invention.

1. A method of modulating a communication signal including: generating afamily of shift sequences having lengths relatively prime usingexponential, logarithmic or index functions and a polynomial or arational function polynomial in i∈

_(p−1) for a finite field

_(p) of prime p; substituting multiple columns of arrays having arelatively prime size with pseudo-noise sequences in a cyclic shiftequal to the shift sequence for the respective column of the arrays togenerate a substituted array; unfolding sequences from the arrays usingChinese Remainder Theorem; and applying the unfolded sequences to acarrier wave of the communication signal to generate a phase modulatedcommunication signal.
 2. The method of claim 1 wherein a shift sequenceof the family has the form: s_(i)=Ag^(2i)+Bg^(i)+C where g is aprimitive root (generator) of Z_(p) and i ranges from 1 to p−1. where A,B, C are elements of the base field Z_(p)
 3. The method of claim 1,including generating one of the following: a (p−1)×p array having (p−1)columns each of length p; a p×(p+1) array with p columns, each of length(p+1) where the index function is used; a p×(p+1) array with p columns,each of length (p+1); the family of shift sequences using a quadraticexponential map of Construction A1 or A2; or the family of shiftsequences using a quadratic discrete logarithm map as an inverse ofConstruction A1 or A2. 4-7. (canceled)
 8. The method of claim 1, whereinthe step of generating the family of shift sequences includes one of thefollowing: using full cycles generated by a rational function map,producing p×(p+1) arrays with p columns, each of length (p+1), whereinthe rational function map is the Family B rational function map; usingknown frequency hopping patterns, time hopping patterns or opticalorthogonal codes; transforming known CDMA families into shift sequences,wherein the known CDMA families are one of the following: small Kasamisequences; large Kasami sequences; No-Kumar sequences. 9-12. (canceled)13. The method of claim 1, wherein the pseudo-noise sequence is over anyalphabet and can be one of the following: a binary or almost binaryLegendre sequence; a ternary or polyphase Legendre sequence; a binary orpolyphase m-sequence; a GMW sequence; a twin prime sequence; a Hallsequence; a Sidelnikov sequence; another low off-peak auto-correlationsequence.
 14. The method of claim 1, wherein the substituted array is inthe form of, or used for the modulation of one of the following: abi-phase array for modulation of a CDMA signal; a multi-target trackingradar signal; a multi-target tracking sonar signal; an ultrasoundsignal; an optical orthogonal code array for modulation of an opticalCDMA signal.
 15. The method of claim 1, including substituting at leastone column of the arrays with a constant column such that thesubstituted array is balanced and has symmetric auto-correlation values,optionally including substituting blank columns in the shift sequence tobalance a CDMA sequence.
 16. The method of claim 1, includingsubstituting multiple rows of the arrays with an array comprising amaximum of one dot per row.
 17. The method of claim 1, includingconstructing groups of families of sequences or arrays by applyinginvariance operations to parent arrays and unfolding the parent or thetransformed arrays using the Chinese Remainder Theorem.
 18. The methodof claim 17, including assigning different groups of users differentfamilies of sequences or arrays.
 19. (canceled)
 20. The method of claim1, wherein the shift sequences are used in their own right for one ofthe following: frequency hopping patterns; time hopping patterns;optical orthogonal codes; sonar sequences.
 21. (canceled)
 22. The methodof claim 1 including increasing linear complexity by using a shiftsequence and a column sequence in composition, optionally includingunfolding the composition into a long sequence using the ChineseRemainder Theorem, so that the resulting multidimensional array can beused in image, audio, video or multimedia watermarking.
 23. The methodof claim 1 including generating a family of multidimensional arraysusing a composition of a family of shift sequences or frequency hoppatterns and a column sequence and modulating the phase of amultiplicity of orthogonal carriers and optionally dual polarization.24. The method of claim 1, wherein the sequences are used as one of thefollowing: an error correcting code; a cryptographic code.
 25. Themethod of claim 1 including converting a column solitary sequence withlow off-peak autocorrelation into a family of longer sequences or arrayswith low off-peak autocorrelation and cross-correlation.
 26. The methodof claim 1 including unfolding a solitary array from Construction A1 orA2 using a degree one polynomial using Chinese Remainder Theorem, andsubstituting rows in a larger array used to produce Family B withcommensurate row length with the unfolded solitary array, resulting in afamily of even larger arrays with good correlation and high linearcomplexity.
 27. The method of claim 26, wherein the substitution isperformed as a cascade or the process is performed recursively. 28.(canceled)
 29. The method of claim 1, wherein the family of sequencesproduced by using the shift sequence to construct arrays which are thenunfolded using Chinese Remainder Theorem has a linear complexity greaterthan 45% of the sequence length, regardless of the sequence length. 30.The method of claim 1, wherein the shift sequence is obtained from anm-sequence, and the substitution column is a ternary or other non-binarypseudo-noise sequence, resulting in a new long pseudo-noise sequenceobtained by Chinese Remainder Theorem from the array.
 31. The method ofclaim 1, wherein a family of shift sequences is obtained by acombination of trace map and discrete logarithm from a singly or doublyperiodic shift sequence produced by an m-sequence.
 32. The method ofclaim 31, where the family thus produced is the small Kasami set, or theNo-Kumar set.
 33. A device for modulating a communication signal, thedevice comprising: a transceiver for receiving and transmitting thesignal; a storage medium storing computer implemented programme codecomponents to generate sequences; and a processor in communication withthe storage medium and the transceiver to execute at least some of thecomputer implemented programme code components to cause: generating afamily of shift sequences or arrays using exponential, logarithmic orindex functions and a polynomial or a rational function polynomial in i∈

_(p−1) for a finite field

_(p) of prime p; substituting multiple columns of the arrays withpseudo-noise sequences in a cyclic shift equal to the shift sequence forthe respective column to generate a substituted array; and applying thesubstituted array, or a sequence unfolded from the array when the arraydimensions are relatively prime, to a carrier wave of the communicationsignal to generate a modulated communication signal.
 34. A device formodulating a communication signal, the device comprising: a transceiverfor receiving and transmitting the signal; a storage medium storingcomputer implemented programme code components to generate sequences;and a processor in communication with the storage medium and thetransceiver to execute at least some of the computer implementedprogramme code components to cause: generating a family of shiftsequences having lengths relatively prime using exponential, logarithmicor index functions and a polynomial or a rational function polynomial ini∈

_(p−1) for a finite field

_(p) of prime p; substituting multiple columns of arrays having arelatively prime size with pseudo-noise sequences in a cyclic shiftequal to the shift sequence for the respective column of the arrays togenerate a substituted array; unfolding sequences from the arrays usingChinese Remainder Theorem; applying the unfolded sequences to a carrierwave of the communication signal to generate a phase modulatedcommunication signal.